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A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if
you know any term xn, you can find the next term xn+1 using the
formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of
this sequence. What do you notice? Calculate a few more terms and
find the squares of the terms. Can you prove that the special
property you notice about this sequence will apply to all the later
terms of the sequence? Write down a formula to give an
approximation to the cube root of a number and test it for the cube
root of 3 and the cube root of 8. How many terms of the sequence do
you have to take before you get the cube root of 8 correct to as
many decimal places as your calculator will give? What happens when
you try this method for fourth roots or fifth roots etc.?

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

Sorted

Stage: 5 Challenge Level:

Why do this problem?

This problem introduces sorting algorithms by encouraging students to explore them using a pack of cards. By performing the algorithms for themselves, we hope students will gain a better understanding of the advantages and limitations of each method.

Possible approach

Each student will need one suit from a pack of cards.
"Shuffle your cards, and then put them in order from Ace to King. Watch how your partner puts their cards in order. Do you both do it the same way? How efficient is your method?"

Either show students each video and invite them to make sense of the algorithm and have a go themselves,Or hand out this worksheet for them to make sense of the algorithms on paper.

"For each of the algorithms, perform it a few times to get a feel for it. Then choose two algorithms and compare them. Which is the quickest? Why? Can you put your cards into a worst-case scenario for each of the algorithms, to make it take as long as possible?"

Give students time to explore these, together with the questions from the problem which are on the worksheet:

On average, which algorithm did you find to be quickest?

What is the 'worst-case scenario' for each algorithm?

How long would it take in the worst case?

Which would you choose if you had to keep the cards in a pile rather than laying them out

Which would you choose if you only had a limited amount of desk space to arrange the cards on?

Then bring the class together to discuss their findings.

Possible extension

Invite students to use pseudocode, or a programming language if they know one, to express the algorithms.

Possible support

Introduce the algorithms one at a time, and then make pairwise comparisons between them.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.